In recent years, the field of theoretical physics has experienced a revival of interest in the exploration of integrable and quantum chaotic properties of physical systems. This workshop aims to bring together researchers from condensed matter, mathematical and high-energy physics communities to facilitate a cross-disciplinary discussion on these rapidly evolving developments.

The workshop will be held between 3 – 9 June, 2018, in Bled, Slovenia.

It is well known that typical pure states in the Hilbert space are (nearly) maximally entangled. In my talk I will discuss, from the perspective of bipartite entanglement entropy, how different are typical eigenstates of physical Hamiltonians from typical states in the Hilbert space.

In the first part, I will present tools to compute the average entanglement entropy of all eigenstates of translationally-invariant quadratic fermionic Hamiltonians, and derive exact bounds [1]. I will prove that (i) if the subsystem volume is a finite fraction of the system volume, then the average entanglement entropy is smaller than the result for typical pure states in the thermodynamic limit (the difference is extensive with system volume), and (ii) in the limit in which the subsystem volume is a vanishing fraction of the system volume, the average entanglement entropy is maximal; i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.

In the second part, I will focus on eigenstates of quantum chaotic many-body Hamiltonians [2]. I will prove that, in a system that is away from half filling and divided in two equal halves, an upper bound for the average entanglement entropy of random pure states with a fixed particle number and normally distributed real coefficients exhibits a deviation from the maximal value that grows with the square root of the volume of the system. Exact numerical results for highly excited eigenstates of a particle number conserving quantum chaotic model indicate that the bound is saturated with increasing system volume.

My talk will present two recently uncovered connections between hydrodynamics and many-body chaos—one at strong and one at weak coupling. I will begin by introducing some relevant concepts from the field of relativistic hydrodynamics, which will be followed by a short discussion of holographic duality. In particular, I will describe how thermal field theory correlation functions can be inferred from the properties of black holes. I will then show how in strongly interacting theories with simple holographic duals both the Lyapunov exponent and the butterfly velocity can be computed from a retarded longitudinal energy-energy two-point function, without the need for computing the OTOC. This makes the connection between holographic many-body chaos and hydrodynamics precise. Finally, by constructing a modified kinetic, Boltzmann-like equation, both phenomenologically and from an OTOC, I will argue that even at weak coupling, hydrodynamic and chaotic properties can be intimately related.

In the limit of small ħ, e.g. short wavelengths, semi-classical theory provides a connection between the spectrum of a quantum mechanical system and the periodic orbits of its classical counter part. While this is well established for systems with few degrees of freedom the extension to many-body systems is challenging as the system dimension N turns into an additional large parameter. For a long chain of interacting kicked tops short time spectral information may be obtained using a duality relation between the temporal and spatial evolution. This allows a direct comparison to the classical periodic orbits of the system. It reveals that this class of system possesses non-isolated orbit manifolds whose contributions to the trace formula are strongly dominant. For large N their presence leads to remarkably strong spectral fluctuations, which are not recovered by conventional semi-classical theory.

It is a well known fact that the spreading of information in lattice quantum systems is not instantaneous, but it rather exists a maximum velocity dictated by the Lieb-Robinson bound.The existence of such a lightcone deeply affects equilibrium properties as well as out-of-equilibrium ones, which have been a subject of outstanding interest in the recent years.In this talk, a novel out-of-equilibrium protocol critically affected by the presence of a maximum velocity is proposed and discussed. Specifically, a one dimensional lattice model is considered, where a localized impurity is suddenly created and then dragged at a constant velocity.Focussing on a simple, but far from trivial, free model the response of the system at late times is analyzed, with emphasis on its transport properties. The finite maximum velocity is responsible for a rich phenomenology, for which exact results are provided. Taking into account the experience acquired so far, more general models are discussed and unpublished results presented, with exact predictions in completely generic (non integrable) one dimensional lattice systems.

In this talk, we shall consider the classical field theory of the Heisenberg ferromagnet, which is the semi-classical limit of the Heisenberg spin-1/2 chain. Our primary motivation to study classical solvable dynamical system is to understand the similarities and differences between integrable quantum systems and their classical counterparts. Unfortunately, there seem to exist no efficient computational framework to achieve this goal. To this end, by starting from first principles and employing the finite-gap algebro-geometric integration technique, we identify a certain scaling procedure of the corresponding degenerate large-genus Riemann surfaces which provides a statistical description for a thermodynamic gas of magnetic solitons. Using simple arguments of semi-classical quantization, we use the classical S-matrix to derive a universal integral dressing equation for the spectral distribution function of soliton excitations which accounts for interactions with a non-trivial many-body vacuum. By lifting the conventional theory of Whitham modulations equations to the thermodynamic setting, we obtain a simple formula for the effective propagation velocity which is, remarkably, in formal agreement with that proposed recently for quantum integrable models.

I’ll give a short review of the recent theoretical progress to explicitly construct non-thermal steady states in quantum systems such as interacting bosons and spin chains. Moreover, I’ll present the recently introduced hydrodynamic description of such non-thermal steady states that allows to study (ballistic) transport properties of many-body systems and to construct non-equilibrium steady states with persistent energy or spin currents and stronger quantum correlations.

In this talk I will argue that novel ordered phases can be expected for interacting quantum systems away from thermal equilibrium. I will start by reporting a set of mean-field results concerning the effects of large bias voltages applied across an half-filled Hubbard chain. As a function of the applied voltage and temperature a rich set of phases can be found that is induced by the interplay between electron-electron interactions and non-equilibrium conditions. Taking a step back, I try to explain why such phases are possible (at least at the mean field level). This will motivate the characterization of the current-carrying steady-state that arises in the middle of a non-interacting metallic wire connected to macroscopic leads. Finally, I will comment on some ongoing work regarding the fate of the Peierls transition in a similar non-equilibrium setup.

I will discuss a general procedure to construct an integrable real–time trotterization of interacting lattice models. As an illustrative example we will consider a spin-$1/2$ chain, with continuous time dynamics described by the isotropic ($XXX$) Heisenberg Hamiltonian. I will derive local conservation laws from an inhomogeneous transfer matrix and construct a boost operator. In the continuous time limit these local charges reduce to the known integrals of motion of the Heisenberg chain.

In a simple Kraus representation I will examine the nonequilibrium setting, where our integrable cellular automaton is driven by stochastic processes at the boundaries.
We will see, how an exact nonequilibrium steady state density matrix can be written in terms of a staggered matrix product ansatz.

This simple trotterization scheme, in particular in the open system framework, could prove to be a useful tool for experimental simulations of the lattice models in terms of trapped ion and atom optics setups.

* Lenart Zadnik, Faculty of Mathematics and Physics, University of Ljubljana

The properties of a Hilbert space may sometimes be usefully illuminated by expressing its states with respect to an overcomplete basis parameterized by the points of a smooth manifold. A prime example of the technique is the Segal-Bargmann representation wherein states of a single-mode bosonic Fock space are expanded in terms of the overcomplete basis of coherent states. The Fock space is then found to be isomorphic to the space of holomorphic functions of a certain finite norm. Furthermore, the creation and annihilation operators, and any function of them, can be expressed as functions of the complex position and complex derivative operators.

In this talk I will present the general theory of embedding a Hilbert space in a suitable space of functions over a smooth manifold that parameterizes an overcomplete basis in the original space. In many cases, operators of theoretical interest in the original space may be mapped onto differential operators on the smooth manifold such that the spectrum of the mapped operator contains the spectrum of the original one. This, in particular, allows the use of calculus and geometrical reasoning when diagonalizing Hamiltonians in the original space.

I will demonstrate the technique on a general d-mode many-body system that may be mapped onto a single-particle problem on the (d-1)-sphere. I will finally review some low-d applications of the formalism, which have found utility in the context of few-site tight-binding Hamiltonians and Bose-Einstein condensates of spinful atoms.